This is a weird aspect of punctuation. Some people believe you can put anything you like inside quotes and it doesn't end the sentence - others believe you can't. There's no "right or wrong" here, according to me, but an examiner may impose his own preferences as rules. Personally I would choose the comma for the first, but I would put it outside of the quotes (inside and outside are both popular styles in the UK). I would need to say more context to decide on the second one, but I'm leaning towards the full stop there - again, outside the quotes, but this is a style thing not a law, according to me.

The second sentence is correct.